Players 1 and 2 each simultaneously choose an integer from the set {1, 2, 3}. If the two players choose the same integer , then player 2 pays to player 1; otherwise, no payment is made. Each player seeks to maximize her expected monetary payoff. There is no pure strategy equilibrium in this game, but there is a mixed strategy equilibrium. Suppose that player 1 plays the mixed strategy (1, 2, 3), where is the probability that integer is chosen, and that player 2 plays the mixed strategy (1, 2, 3). Suppose that this strategy profile is a Nash equilibrium.

a) Show that p_n >0and q_n >0 for any n {1,2,3}.

b) Find a mixed strategy Nash equilibrium of this game.